LMFDB - Embedded newform 712.1.y.a.691.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [712,1,Mod(99,712)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(712, base_ring=CyclotomicField(44))

chi = DirichletCharacter(H, H._module([22, 22, 43]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("712.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 712 = 2^{3} \cdot 89 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	712.y (of order $44$, degree $20$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.355334288995$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{44})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{44}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{44} - \cdots)$

Embedding invariants

Embedding label			691.1
Root			$-0.989821 - 0.142315i$ of defining polynomial
Character	$\chi$	$=$	712.691
Dual form			712.1.y.a.339.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.959493 - 0.281733i) q^{2} +(0.254771 + 1.17116i) q^{3} +(0.841254 - 0.540641i) q^{4} +(0.574406 + 1.05195i) q^{6} +(0.654861 - 0.755750i) q^{8} +(-0.397086 + 0.181343i) q^{9} +O(q^{10})$	\(q+(0.959493 - 0.281733i) q^{2} +(0.254771 + 1.17116i) q^{3} +(0.841254 - 0.540641i) q^{4} +(0.574406 + 1.05195i) q^{6} +(0.654861 - 0.755750i) q^{8} +(-0.397086 + 0.181343i) q^{9} +(-0.989821 - 1.14231i) q^{11} +(0.847507 + 0.847507i) q^{12} +(0.415415 - 0.909632i) q^{16} +(-0.540641 + 1.84125i) q^{17} +(-0.329911 + 0.285870i) q^{18} +(-1.86912 - 0.697148i) q^{19} +(-1.27155 - 0.817178i) q^{22} +(1.05195 + 0.574406i) q^{24} +(0.142315 + 0.989821i) q^{25} +(0.404719 + 0.540641i) q^{27} +(0.142315 - 0.989821i) q^{32} +(1.08566 - 1.45027i) q^{33} +1.91899i q^{34} +(-0.236009 + 0.367237i) q^{36} +(-1.98982 - 0.142315i) q^{38} +(-1.38189 - 0.300613i) q^{41} +(0.125226 - 1.75089i) q^{43} +(-1.45027 - 0.425839i) q^{44} +(1.17116 + 0.254771i) q^{48} +(0.989821 - 0.142315i) q^{49} +(0.415415 + 0.909632i) q^{50} +(-2.29415 - 0.164081i) q^{51} +(0.540641 + 0.404719i) q^{54} +(0.340275 - 2.36667i) q^{57} +(-0.203743 + 0.936593i) q^{59} +(-0.142315 - 0.989821i) q^{64} +(0.633095 - 1.69739i) q^{66} +(1.53046 + 0.983568i) q^{67} +(0.540641 + 1.84125i) q^{68} +(-0.122986 + 0.418852i) q^{72} +(0.822373 - 1.80075i) q^{73} +(-1.12299 + 0.418852i) q^{75} +(-1.94931 + 0.424047i) q^{76} +(-0.815938 + 0.941643i) q^{81} +(-1.41061 + 0.100889i) q^{82} +(0.203743 + 0.373128i) q^{83} +(-0.373128 - 1.71524i) q^{86} -1.51150 q^{88} +(0.142315 - 0.989821i) q^{89} +(1.19550 - 0.0855040i) q^{96} +(-0.544078 + 0.627899i) q^{97} +(0.909632 - 0.415415i) q^{98} +(0.600195 + 0.274100i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8}+O(q^{10}) $ 20 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 2 * q^8	$ 20 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8} - 2 q^{12} - 2 q^{16} - 2 q^{19} + 2 q^{24} + 2 q^{25} - 22 q^{27} + 2 q^{32} - 20 q^{38} + 2 q^{41} + 2 q^{43} - 2 q^{48} - 2 q^{50} + 4 q^{51} - 4 q^{57} - 2 q^{59} - 2 q^{64} + 22 q^{72} + 2 q^{75} - 2 q^{76} - 2 q^{81} - 2 q^{82} + 2 q^{83} - 2 q^{86} + 2 q^{89} + 2 q^{96} - 4 q^{97}+O(q^{100}) $ 20 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 2 * q^8 - 2 * q^12 - 2 * q^16 - 2 * q^19 + 2 * q^24 + 2 * q^25 - 22 * q^27 + 2 * q^32 - 20 * q^38 + 2 * q^41 + 2 * q^43 - 2 * q^48 - 2 * q^50 + 4 * q^51 - 4 * q^57 - 2 * q^59 - 2 * q^64 + 22 * q^72 + 2 * q^75 - 2 * q^76 - 2 * q^81 - 2 * q^82 + 2 * q^83 - 2 * q^86 + 2 * q^89 + 2 * q^96 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/712\mathbb{Z}\right)^\times$.

$n$	$357$	$535$	$537$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{19}{44}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.959493	−	0.281733i	0.959493	−	0.281733i
$2$	0.959493	−	0.281733i	0.959493	−	0.281733i
$3$	0.254771	+	1.17116i	0.254771	+	1.17116i	0.909632	+	0.415415i	$0.136364\pi$
$3$	0.254771	+	1.17116i	0.254771	+	1.17116i	−0.654861	+	0.755750i	$0.727273\pi$
$4$	0.841254	−	0.540641i	0.841254	−	0.540641i
$5$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$5$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$6$	0.574406	+	1.05195i	0.574406	+	1.05195i
$7$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$7$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$8$	0.654861	−	0.755750i	0.654861	−	0.755750i
$9$	−0.397086	+	0.181343i	−0.397086	+	0.181343i
$10$		0			0
$11$	−0.989821	−	1.14231i	−0.989821	−	1.14231i	−0.989821	−	0.142315i	$-0.954545\pi$
$11$	−0.989821	−	1.14231i	−0.989821	−	1.14231i		−	1.00000i	$-0.5\pi$
$12$	0.847507	+	0.847507i	0.847507	+	0.847507i
$13$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$13$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$14$		0			0
$15$		0			0
$16$	0.415415	−	0.909632i	0.415415	−	0.909632i
$17$	−0.540641	+	1.84125i	−0.540641	+	1.84125i			1.00000i	$0.5\pi$
$17$	−0.540641	+	1.84125i	−0.540641	+	1.84125i	−0.540641	+	0.841254i	$0.681818\pi$
$18$	−0.329911	+	0.285870i	−0.329911	+	0.285870i
$19$	−1.86912	−	0.697148i	−1.86912	−	0.697148i	−0.959493	−	0.281733i	$-0.909091\pi$
$19$	−1.86912	−	0.697148i	−1.86912	−	0.697148i	−0.909632	−	0.415415i	$-0.863636\pi$
$20$		0			0
$21$		0			0
$22$	−1.27155	−	0.817178i	−1.27155	−	0.817178i
$23$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$23$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$24$	1.05195	+	0.574406i	1.05195	+	0.574406i
$25$	0.142315	+	0.989821i	0.142315	+	0.989821i
$26$		0			0
$27$	0.404719	+	0.540641i	0.404719	+	0.540641i
$28$		0			0
$29$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$29$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$30$		0			0
$31$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$31$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$32$	0.142315	−	0.989821i	0.142315	−	0.989821i
$33$	1.08566	−	1.45027i	1.08566	−	1.45027i
$34$			1.91899i			1.91899i
$35$		0			0
$36$	−0.236009	+	0.367237i	−0.236009	+	0.367237i
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	−1.98982	−	0.142315i	−1.98982	−	0.142315i
$39$		0			0
$40$		0			0
$41$	−1.38189	−	0.300613i	−1.38189	−	0.300613i	−0.540641	−	0.841254i	$-0.681818\pi$
$41$	−1.38189	−	0.300613i	−1.38189	−	0.300613i	−0.841254	+	0.540641i	$0.818182\pi$
$42$		0			0
$43$	0.125226	−	1.75089i	0.125226	−	1.75089i	−0.415415	−	0.909632i	$-0.636364\pi$
$43$	0.125226	−	1.75089i	0.125226	−	1.75089i	0.540641	−	0.841254i	$-0.318182\pi$
$44$	−1.45027	−	0.425839i	−1.45027	−	0.425839i
$45$		0			0
$46$		0			0
$47$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$47$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$48$	1.17116	+	0.254771i	1.17116	+	0.254771i
$49$	0.989821	−	0.142315i	0.989821	−	0.142315i
$50$	0.415415	+	0.909632i	0.415415	+	0.909632i
$51$	−2.29415	−	0.164081i	−2.29415	−	0.164081i
$52$		0			0
$53$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$53$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$54$	0.540641	+	0.404719i	0.540641	+	0.404719i
$55$		0			0
$56$		0			0
$57$	0.340275	−	2.36667i	0.340275	−	2.36667i
$58$		0			0
$59$	−0.203743	+	0.936593i	−0.203743	+	0.936593i	0.755750	+	0.654861i	$0.227273\pi$
$59$	−0.203743	+	0.936593i	−0.203743	+	0.936593i	−0.959493	+	0.281733i	$0.909091\pi$
$60$		0			0
$61$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$61$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$62$		0			0
$63$		0			0
$64$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$65$		0			0
$66$	0.633095	−	1.69739i	0.633095	−	1.69739i
$67$	1.53046	+	0.983568i	1.53046	+	0.983568i	0.989821	+	0.142315i	$0.0454545\pi$
$67$	1.53046	+	0.983568i	1.53046	+	0.983568i	0.540641	+	0.841254i	$0.318182\pi$
$68$	0.540641	+	1.84125i	0.540641	+	1.84125i
$69$		0			0
$70$		0			0
$71$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$71$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$72$	−0.122986	+	0.418852i	−0.122986	+	0.418852i
$73$	0.822373	−	1.80075i	0.822373	−	1.80075i	0.281733	−	0.959493i	$-0.409091\pi$
$73$	0.822373	−	1.80075i	0.822373	−	1.80075i	0.540641	−	0.841254i	$-0.318182\pi$
$74$		0			0
$75$	−1.12299	+	0.418852i	−1.12299	+	0.418852i
$76$	−1.94931	+	0.424047i	−1.94931	+	0.424047i
$77$		0			0
$78$		0			0
$79$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$79$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$80$		0			0
$81$	−0.815938	+	0.941643i	−0.815938	+	0.941643i
$82$	−1.41061	+	0.100889i	−1.41061	+	0.100889i
$83$	0.203743	+	0.373128i	0.203743	+	0.373128i	0.959493	−	0.281733i	$-0.0909091\pi$
$83$	0.203743	+	0.373128i	0.203743	+	0.373128i	−0.755750	+	0.654861i	$0.772727\pi$
$84$		0			0
$85$		0			0
$86$	−0.373128	−	1.71524i	−0.373128	−	1.71524i
$87$		0			0
$88$	−1.51150			−1.51150
$89$	0.142315	−	0.989821i	0.142315	−	0.989821i
$89$	0.142315	−	0.989821i	0.142315	−	0.989821i
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	1.19550	−	0.0855040i	1.19550	−	0.0855040i
$97$	−0.544078	+	0.627899i	−0.544078	+	0.627899i	−0.959493	−	0.281733i	$-0.909091\pi$
$97$	−0.544078	+	0.627899i	−0.544078	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$98$	0.909632	−	0.415415i	0.909632	−	0.415415i
$99$	0.600195	+	0.274100i	0.600195	+	0.274100i
$100$	0.654861	+	0.755750i	0.654861	+	0.755750i
$101$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$101$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$102$	−2.24745	+	0.488902i	−2.24745	+	0.488902i
$103$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$103$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$107$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$108$	0.632763	+	0.236009i	0.632763	+	0.236009i
$109$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$109$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.373128	+	0.203743i	0.373128	+	0.203743i	0.654861	−	0.755750i	$-0.272727\pi$
$113$	0.373128	+	0.203743i	0.373128	+	0.203743i	−0.281733	+	0.959493i	$0.590909\pi$
$114$	−0.340275	−	2.36667i	−0.340275	−	2.36667i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	0.0683785	+	0.956056i	0.0683785	+	0.956056i
$119$		0			0
$120$		0			0
$121$	−0.182822	+	1.27155i	−0.182822	+	1.27155i
$122$		0			0
$123$		−	1.69501i		−	1.69501i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$127$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$128$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$129$	2.08248	−	0.299415i	2.08248	−	0.299415i
$130$		0			0
$131$	0.153882	+	0.239446i	0.153882	+	0.239446i	0.909632	−	0.415415i	$-0.136364\pi$
$131$	0.153882	+	0.239446i	0.153882	+	0.239446i	−0.755750	+	0.654861i	$0.772727\pi$
$132$	0.129239	−	1.80700i	0.129239	−	1.80700i
$133$		0			0
$134$	1.74557	+	0.512546i	1.74557	+	0.512546i
$135$		0			0
$136$	1.03748	+	1.61435i	1.03748	+	1.61435i
$137$	−0.936593	−	0.203743i	−0.936593	−	0.203743i	−0.281733	−	0.959493i	$-0.590909\pi$
$137$	−0.936593	−	0.203743i	−0.936593	−	0.203743i	−0.654861	+	0.755750i	$0.727273\pi$
$138$		0			0
$139$	0.544078	+	1.19136i	0.544078	+	1.19136i	0.959493	+	0.281733i	$0.0909091\pi$
$139$	0.544078	+	1.19136i	0.544078	+	1.19136i	−0.415415	+	0.909632i	$0.636364\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			0.436535i			0.436535i
$145$		0			0
$146$	0.281733	−	1.95949i	0.281733	−	1.95949i
$147$	0.418852	+	1.12299i	0.418852	+	1.12299i
$148$		0			0
$149$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$149$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$150$	−0.959493	+	0.718267i	−0.959493	+	0.718267i
$151$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$151$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$152$	−1.75089	+	0.956056i	−1.75089	+	0.956056i
$153$	−0.119218	−	0.829178i	−0.119218	−	0.829178i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$157$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.517596	+	1.13338i	−0.517596	+	1.13338i
$163$	−0.767317	+	1.40524i	−0.767317	+	1.40524i	0.142315	+	0.989821i	$0.454545\pi$
$163$	−0.767317	+	1.40524i	−0.767317	+	1.40524i	−0.909632	+	0.415415i	$0.863636\pi$
$164$	−1.32505	+	0.494217i	−1.32505	+	0.494217i
$165$		0			0
$166$	0.300613	+	0.300613i	0.300613	+	0.300613i
$167$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$167$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$168$		0			0
$169$	0.909632	−	0.415415i	0.909632	−	0.415415i
$170$		0			0
$171$	0.868626	−	0.0621254i	0.868626	−	0.0621254i
$172$	−0.841254	−	1.54064i	−0.841254	−	1.54064i
$173$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$173$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$174$		0			0
$175$		0			0
$176$	−1.45027	+	0.425839i	−1.45027	+	0.425839i
$177$	−1.14881			−1.14881
$178$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$179$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$179$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$180$		0			0
$181$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$181$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	2.63843	−	1.20493i	2.63843	−	1.20493i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$191$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$192$	1.12299	−	0.418852i	1.12299	−	0.418852i
$193$	0.334961	−	0.613435i	0.334961	−	0.613435i	−0.654861	−	0.755750i	$-0.727273\pi$
$193$	0.334961	−	0.613435i	0.334961	−	0.613435i	0.989821	+	0.142315i	$0.0454545\pi$
$194$	−0.345139	+	0.755750i	−0.345139	+	0.755750i
$195$		0			0
$196$	0.755750	−	0.654861i	0.755750	−	0.654861i
$197$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$197$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$198$	0.653106	+	0.0939025i	0.653106	+	0.0939025i
$199$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$199$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$200$	0.841254	+	0.540641i	0.841254	+	0.540641i
$201$	−0.762003	+	2.04301i	−0.762003	+	2.04301i
$202$		0			0
$203$		0			0
$204$	−2.01867	+	1.10228i	−2.01867	+	1.10228i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.05374	+	2.82518i	1.05374	+	2.82518i
$210$		0			0
$211$	−1.05195	+	1.40524i	−1.05195	+	1.40524i	−0.142315	+	0.989821i	$0.545455\pi$
$211$	−1.05195	+	1.40524i	−1.05195	+	1.40524i	−0.909632	+	0.415415i	$0.863636\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$	0.673623	+	0.0481785i	0.673623	+	0.0481785i
$217$		0			0
$218$		0			0
$219$	2.31849	+	0.504356i	2.31849	+	0.504356i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$223$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$224$		0			0
$225$	−0.236009	−	0.367237i	−0.236009	−	0.367237i
$226$	0.415415	+	0.0903680i	0.415415	+	0.0903680i
$227$	−1.95949	+	0.281733i	−1.95949	+	0.281733i	−0.959493	+	0.281733i	$0.909091\pi$
$227$	−1.95949	+	0.281733i	−1.95949	+	0.281733i	−1.00000			$\pi$
$228$	−0.993259	−	2.17493i	−0.993259	−	2.17493i
$229$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$229$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1.68251i			1.68251i	0.540641	+	0.841254i	$0.318182\pi$
$233$			1.68251i			1.68251i	−0.540641	+	0.841254i	$0.681818\pi$
$234$		0			0
$235$		0			0
$236$	0.334961	+	0.898064i	0.334961	+	0.898064i
$237$		0			0
$238$		0			0
$239$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$239$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$240$		0			0
$241$	1.71524	−	0.936593i	1.71524	−	0.936593i	0.755750	−	0.654861i	$-0.227273\pi$
$241$	1.71524	−	0.936593i	1.71524	−	0.936593i	0.959493	−	0.281733i	$-0.0909091\pi$
$242$	0.182822	+	1.27155i	0.182822	+	1.27155i
$243$	−0.717961	−	0.392036i	−0.717961	−	0.392036i
$244$		0			0
$245$		0			0
$246$	−0.477540	−	1.62635i	−0.477540	−	1.62635i
$247$		0			0
$248$		0			0
$249$	−0.385087	+	0.333679i	−0.385087	+	0.333679i
$250$		0			0
$251$	0.698939	−	1.53046i	0.698939	−	1.53046i	−0.142315	−	0.989821i	$-0.545455\pi$
$251$	0.698939	−	1.53046i	0.698939	−	1.53046i	0.841254	−	0.540641i	$-0.181818\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$257$	−1.19136	−	0.544078i	−1.19136	−	0.544078i	−0.281733	−	0.959493i	$-0.590909\pi$
$257$	−1.19136	−	0.544078i	−1.19136	−	0.544078i	−0.909632	+	0.415415i	$0.863636\pi$
$258$	1.91377	−	0.873989i	1.91377	−	0.873989i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.215109	+	0.186393i	0.215109	+	0.186393i
$263$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$263$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$264$	−0.385087	−	1.77021i	−0.385087	−	1.77021i
$265$		0			0
$266$		0			0
$267$	1.19550	−	0.0855040i	1.19550	−	0.0855040i
$268$	1.81926			1.81926
$269$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$269$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$270$		0			0
$271$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$271$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$272$	1.45027	+	1.25667i	1.45027	+	1.25667i
$273$		0			0
$274$	−0.956056	+	0.0683785i	−0.956056	+	0.0683785i
$275$	0.989821	−	1.14231i	0.989821	−	1.14231i
$276$		0			0
$277$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$277$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$278$	0.857685	+	0.989821i	0.857685	+	0.989821i
$279$		0			0
$280$		0			0
$281$	−0.133682	+	0.0498610i	−0.133682	+	0.0498610i	−0.415415	−	0.909632i	$-0.636364\pi$
$281$	−0.133682	+	0.0498610i	−0.133682	+	0.0498610i	0.281733	+	0.959493i	$0.409091\pi$
$282$		0			0
$283$	0.449181	−	0.983568i	0.449181	−	0.983568i	−0.540641	−	0.841254i	$-0.681818\pi$
$283$	0.449181	−	0.983568i	0.449181	−	0.983568i	0.989821	−	0.142315i	$-0.0454545\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.122986	+	0.418852i	0.122986	+	0.418852i
$289$	−2.25667	−	1.45027i	−2.25667	−	1.45027i
$290$		0			0
$291$	−0.873989	−	0.477234i	−0.873989	−	0.477234i
$292$	−0.281733	−	1.95949i	−0.281733	−	1.95949i
$293$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$293$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$294$	0.718267	+	0.959493i	0.718267	+	0.959493i
$295$		0			0
$296$		0			0
$297$	0.216983	−	0.997454i	0.216983	−	0.997454i
$298$		0			0
$299$		0			0
$300$	−0.718267	+	0.959493i	−0.718267	+	0.959493i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−1.41061	+	1.41061i	−1.41061	+	1.41061i
$305$		0			0
$306$	−0.347995	−	0.762003i	−0.347995	−	0.762003i
$307$	1.66538	−	0.239446i	1.66538	−	0.239446i	0.755750	−	0.654861i	$-0.227273\pi$
$307$	1.66538	−	0.239446i	1.66538	−	0.239446i	0.909632	+	0.415415i	$0.136364\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$311$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$312$		0			0
$313$	0.0498610	−	0.697148i	0.0498610	−	0.697148i	−0.909632	−	0.415415i	$-0.863636\pi$
$313$	0.0498610	−	0.697148i	0.0498610	−	0.697148i	0.959493	−	0.281733i	$-0.0909091\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$317$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.29415	−	3.06463i	2.29415	−	3.06463i
$324$	−0.177320	+	1.23329i	−0.177320	+	1.23329i
$325$		0			0
$326$	−0.340335	+	1.56449i	−0.340335	+	1.56449i
$327$		0			0
$328$	−1.13214	+	0.847507i	−1.13214	+	0.847507i
$329$		0			0
$330$		0			0
$331$	−0.186393	−	1.29639i	−0.186393	−	1.29639i	−0.841254	−	0.540641i	$-0.818182\pi$
$331$	−0.186393	−	1.29639i	−0.186393	−	1.29639i	0.654861	−	0.755750i	$-0.272727\pi$
$332$	0.373128	+	0.203743i	0.373128	+	0.203743i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.133682	−	0.0498610i	−0.133682	−	0.0498610i	0.281733	−	0.959493i	$-0.409091\pi$
$337$	−0.133682	−	0.0498610i	−0.133682	−	0.0498610i	−0.415415	+	0.909632i	$0.636364\pi$
$338$	0.755750	−	0.654861i	0.755750	−	0.654861i
$339$	−0.143555	+	0.488902i	−0.143555	+	0.488902i
$340$		0			0
$341$		0			0
$342$	0.815938	−	0.304329i	0.815938	−	0.304329i
$343$		0			0
$344$	−1.24123	−	1.24123i	−1.24123	−	1.24123i
$345$		0			0
$346$		0			0
$347$	−0.983568	+	0.449181i	−0.983568	+	0.449181i	−0.841254	−	0.540641i	$-0.818182\pi$
$347$	−0.983568	+	0.449181i	−0.983568	+	0.449181i	−0.142315	+	0.989821i	$0.545455\pi$
$348$		0			0
$349$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$349$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$350$		0			0
$351$		0			0
$352$	−1.27155	+	0.817178i	−1.27155	+	0.817178i
$353$	0.148568	+	0.682956i	0.148568	+	0.682956i	0.989821	+	0.142315i	$0.0454545\pi$
$353$	0.148568	+	0.682956i	0.148568	+	0.682956i	−0.841254	+	0.540641i	$0.818182\pi$
$354$	−1.10228	+	0.323658i	−1.10228	+	0.323658i
$355$		0			0
$356$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$357$		0			0
$358$	0.797176	−	0.234072i	0.797176	−	0.234072i
$359$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$359$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$360$		0			0
$361$	2.25186	+	1.95125i	2.25186	+	1.95125i
$362$		0			0
$363$	−1.53578	+	0.109841i	−1.53578	+	0.109841i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$367$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$368$		0			0
$369$	0.603245	−	0.131228i	0.603245	−	0.131228i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$373$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$374$	2.19209	−	1.89945i	2.19209	−	1.89945i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.654861	+	1.75575i	−0.654861	+	1.75575i			1.00000i	$0.5\pi$
$379$	−0.654861	+	1.75575i	−0.654861	+	1.75575i	−0.654861	+	0.755750i	$0.727273\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$383$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$384$	0.959493	−	0.718267i	0.959493	−	0.718267i
$385$		0			0
$386$	0.148568	−	0.682956i	0.148568	−	0.682956i
$387$	0.267786	+	0.717961i	0.267786	+	0.717961i
$388$	−0.118239	+	0.822373i	−0.118239	+	0.822373i
$389$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$389$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$390$		0			0
$391$		0			0
$392$	0.540641	−	0.841254i	0.540641	−	0.841254i
$393$	−0.241226	+	0.241226i	−0.241226	+	0.241226i
$394$		0			0
$395$		0			0
$396$	0.653106	−	0.0939025i	0.653106	−	0.0939025i
$397$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$397$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$398$		0			0
$399$		0			0
$400$	0.959493	+	0.281733i	0.959493	+	0.281733i
$401$	−1.74557	−	0.512546i	−1.74557	−	0.512546i	−0.755750	−	0.654861i	$-0.772727\pi$
$401$	−1.74557	−	0.512546i	−1.74557	−	0.512546i	−0.989821	+	0.142315i	$0.954545\pi$
$402$	−0.155554	+	2.17493i	−0.155554	+	2.17493i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−1.62635	+	1.62635i	−1.62635	+	1.62635i
$409$	−0.449181	+	0.698939i	−0.449181	+	0.698939i	−0.989821	−	0.142315i	$-0.954545\pi$
$409$	−0.449181	+	0.698939i	−0.449181	+	0.698939i	0.540641	+	0.841254i	$0.318182\pi$
$410$		0			0
$411$		−	1.14881i		−	1.14881i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.25667	+	0.940730i	−1.25667	+	0.940730i
$418$	1.80700	+	2.41387i	1.80700	+	2.41387i
$419$	−0.613435	+	0.334961i	−0.613435	+	0.334961i	−0.755750	−	0.654861i	$-0.772727\pi$
$419$	−0.613435	+	0.334961i	−0.613435	+	0.334961i	0.142315	+	0.989821i	$0.454545\pi$
$420$		0			0
$421$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$421$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$422$	−0.613435	+	1.64468i	−0.613435	+	1.64468i
$423$		0			0
$424$		0			0
$425$	−1.89945	−	0.273100i	−1.89945	−	0.273100i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$431$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$432$	0.659910	−	0.143555i	0.659910	−	0.143555i
$433$	−1.13214	−	1.13214i	−1.13214	−	1.13214i	−0.989821	−	0.142315i	$-0.954545\pi$
$433$	−1.13214	−	1.13214i	−1.13214	−	1.13214i	−0.142315	−	0.989821i	$-0.545455\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	2.36667	−	0.169267i	2.36667	−	0.169267i
$439$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$439$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$440$		0			0
$441$	−0.367237	+	0.236009i	−0.367237	+	0.236009i
$442$		0			0
$443$	1.03748	−	0.304632i	1.03748	−	0.304632i	0.281733	−	0.959493i	$-0.409091\pi$
$443$	1.03748	−	0.304632i	1.03748	−	0.304632i	0.755750	+	0.654861i	$0.227273\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.909632	+	0.584585i	−0.909632	+	0.584585i	−0.909632	−	0.415415i	$-0.863636\pi$
$449$	−0.909632	+	0.584585i	−0.909632	+	0.584585i			1.00000i	$0.5\pi$
$450$	−0.329911	−	0.285870i	−0.329911	−	0.285870i
$451$	1.02443	+	1.87611i	1.02443	+	1.87611i
$452$	0.424047	−	0.0303285i	0.424047	−	0.0303285i
$453$		0			0
$454$	−1.80075	+	0.822373i	−1.80075	+	0.822373i
$455$		0			0
$456$	−1.56577	−	1.80700i	−1.56577	−	1.80700i
$457$	1.32505	+	1.32505i	1.32505	+	1.32505i	0.909632	+	0.415415i	$0.136364\pi$
$457$	1.32505	+	1.32505i	1.32505	+	1.32505i	0.415415	+	0.909632i	$0.363636\pi$
$458$		0			0
$459$	−1.21426	+	0.452897i	−1.21426	+	0.452897i
$460$		0			0
$461$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$461$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$462$		0			0
$463$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$463$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$464$		0			0
$465$		0			0
$466$	0.474017	+	1.61435i	0.474017	+	1.61435i
$467$	1.61435	+	1.03748i	1.61435	+	1.03748i	0.959493	+	0.281733i	$0.0909091\pi$
$467$	1.61435	+	1.03748i	1.61435	+	1.03748i	0.654861	+	0.755750i	$0.272727\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.574406	+	0.767317i	0.574406	+	0.767317i
$473$	−2.12401	+	1.59002i	−2.12401	+	1.59002i
$474$		0			0
$475$	0.424047	−	1.94931i	0.424047	−	1.94931i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	1.38189	−	1.38189i	1.38189	−	1.38189i
$483$		0			0
$484$	0.533654	+	1.16854i	0.533654	+	1.16854i
$485$		0			0
$486$	−0.799328	−	0.173883i	−0.799328	−	0.173883i
$487$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$487$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$488$		0			0
$489$	−1.84125	−	0.540641i	−1.84125	−	0.540641i
$490$		0			0
$491$	0.0101786	−	0.142315i	0.0101786	−	0.142315i	−0.989821	−	0.142315i	$-0.954545\pi$
$491$	0.0101786	−	0.142315i	0.0101786	−	0.142315i	1.00000			$0$
$492$	−0.916393	−	1.42594i	−0.916393	−	1.42594i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−0.275479	+	0.428654i	−0.275479	+	0.428654i
$499$	−1.56449	−	1.17116i	−1.56449	−	1.17116i	−0.909632	−	0.415415i	$-0.863636\pi$
$499$	−1.56449	−	1.17116i	−1.56449	−	1.17116i	−0.654861	−	0.755750i	$-0.727273\pi$
$500$		0			0
$501$		0			0
$502$	0.239446	−	1.66538i	0.239446	−	1.66538i
$503$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$503$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.718267	+	0.959493i	0.718267	+	0.959493i
$508$		0			0
$509$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$509$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$510$		0			0
$511$		0			0
$512$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$513$	−0.379563	−	1.29267i	−0.379563	−	1.29267i
$514$	−1.29639	−	0.186393i	−1.29639	−	0.186393i
$515$		0			0
$516$	1.59002	−	1.37776i	1.59002	−	1.37776i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.71524	−	0.373128i	1.71524	−	0.373128i	0.755750	−	0.654861i	$-0.227273\pi$
$521$	1.71524	−	0.373128i	1.71524	−	0.373128i	0.959493	+	0.281733i	$0.0909091\pi$
$522$		0			0
$523$	−1.29639	−	1.49611i	−1.29639	−	1.49611i	−0.755750	−	0.654861i	$-0.772727\pi$
$523$	−1.29639	−	1.49611i	−1.29639	−	1.49611i	−0.540641	−	0.841254i	$-0.681818\pi$
$524$	0.258908	+	0.118239i	0.258908	+	0.118239i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−0.868215	−	1.59002i	−0.868215	−	1.59002i
$529$	−0.755750	−	0.654861i	−0.755750	−	0.654861i
$530$		0			0
$531$	−0.0889411	−	0.408856i	−0.0889411	−	0.408856i
$532$		0			0
$533$		0			0
$534$	1.12299	−	0.418852i	1.12299	−	0.418852i
$535$		0			0
$536$	1.74557	−	0.512546i	1.74557	−	0.512546i
$537$	0.211672	+	0.973039i	0.211672	+	0.973039i
$538$		0			0
$539$	−1.14231	−	0.989821i	−1.14231	−	0.989821i
$540$		0			0
$541$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$541$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$542$		0			0
$543$		0			0
$544$	1.74557	+	0.797176i	1.74557	+	0.797176i
$545$		0			0
$546$		0			0
$547$	0.139418	−	0.0303285i	0.139418	−	0.0303285i	−0.142315	−	0.989821i	$-0.545455\pi$
$547$	0.139418	−	0.0303285i	0.139418	−	0.0303285i	0.281733	+	0.959493i	$0.409091\pi$
$548$	−0.898064	+	0.334961i	−0.898064	+	0.334961i
$549$		0			0
$550$	0.627899	−	1.37491i	0.627899	−	1.37491i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	1.10181	+	0.708089i	1.10181	+	0.708089i
$557$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$557$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	2.08337	+	2.78305i	2.08337	+	2.78305i
$562$	−0.114220	+	0.0855040i	−0.114220	+	0.0855040i
$563$	0.0101786	+	0.142315i	0.0101786	+	0.142315i	1.00000			$0$
$563$	0.0101786	+	0.142315i	0.0101786	+	0.142315i	−0.989821	+	0.142315i	$0.954545\pi$
$564$		0			0
$565$		0			0
$566$	0.153882	−	1.07028i	0.153882	−	1.07028i
$567$		0			0
$568$		0			0
$569$	−1.50013	−	1.12299i	−1.50013	−	1.12299i	−0.959493	−	0.281733i	$-0.909091\pi$
$569$	−1.50013	−	1.12299i	−1.50013	−	1.12299i	−0.540641	−	0.841254i	$-0.681818\pi$
$570$		0			0
$571$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$571$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.236009	+	0.367237i	0.236009	+	0.367237i
$577$	0.142315	−	1.98982i	0.142315	−	1.98982i		−	1.00000i	$-0.5\pi$
$577$	0.142315	−	1.98982i	0.142315	−	1.98982i	0.142315	−	0.989821i	$-0.454545\pi$
$578$	−2.57385	−	0.755750i	−2.57385	−	0.755750i
$579$	0.803771	+	0.236009i	0.803771	+	0.236009i
$580$		0			0
$581$		0			0
$582$	−0.973039	−	0.211672i	−0.973039	−	0.211672i
$583$		0			0
$584$	−0.822373	−	1.80075i	−0.822373	−	1.80075i
$585$		0			0
$586$		0			0
$587$	−0.708089	+	1.10181i	−0.708089	+	1.10181i	0.281733	+	0.959493i	$0.409091\pi$
$587$	−0.708089	+	1.10181i	−0.708089	+	1.10181i	−0.989821	+	0.142315i	$0.954545\pi$
$588$	0.959493	+	0.718267i	0.959493	+	0.718267i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.398326	−	1.83107i	0.398326	−	1.83107i	−0.142315	−	0.989821i	$-0.545455\pi$
$593$	0.398326	−	1.83107i	0.398326	−	1.83107i	0.540641	−	0.841254i	$-0.318182\pi$
$594$	−0.0728218	−	1.01818i	−0.0728218	−	1.01818i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$599$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$600$	−0.418852	+	1.12299i	−0.418852	+	1.12299i
$601$	−0.239446	−	0.153882i	−0.239446	−	0.153882i	0.415415	−	0.909632i	$-0.363636\pi$
$601$	−0.239446	−	0.153882i	−0.239446	−	0.153882i	−0.654861	+	0.755750i	$0.727273\pi$
$602$		0			0
$603$	−0.786089	−	0.113022i	−0.786089	−	0.113022i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$607$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$608$	−0.956056	+	1.75089i	−0.956056	+	1.75089i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.548580	−	0.633095i	−0.548580	−	0.633095i
$613$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$613$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$614$	1.53046	−	0.698939i	1.53046	−	0.698939i
$615$		0			0
$616$		0			0
$617$	0.767317	+	1.40524i	0.767317	+	1.40524i	0.909632	+	0.415415i	$0.136364\pi$
$617$	0.767317	+	1.40524i	0.767317	+	1.40524i	−0.142315	+	0.989821i	$0.545455\pi$
$618$		0			0
$619$	0.239446	−	0.153882i	0.239446	−	0.153882i	−0.415415	−	0.909632i	$-0.636364\pi$
$619$	0.239446	−	0.153882i	0.239446	−	0.153882i	0.654861	+	0.755750i	$0.272727\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$626$	−0.148568	−	0.682956i	−0.148568	−	0.682956i
$627$	−3.04029	+	1.95388i	−3.04029	+	1.95388i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$631$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$632$		0			0
$633$	−1.91377	−	0.873989i	−1.91377	−	0.873989i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.215109	+	0.186393i	−0.215109	+	0.186393i	−0.755750	−	0.654861i	$-0.772727\pi$
$641$	−0.215109	+	0.186393i	−0.215109	+	0.186393i	0.540641	+	0.841254i	$0.318182\pi$
$642$		0			0
$643$	−0.822373	−	0.118239i	−0.822373	−	0.118239i	−0.281733	−	0.959493i	$-0.590909\pi$
$643$	−0.822373	−	0.118239i	−0.822373	−	0.118239i	−0.540641	+	0.841254i	$0.681818\pi$
$644$		0			0
$645$		0			0
$646$	1.33782	−	3.58682i	1.33782	−	3.58682i
$647$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$647$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$648$	0.177320	+	1.23329i	0.177320	+	1.23329i
$649$	1.27155	−	0.694321i	1.27155	−	0.694321i
$650$		0			0
$651$		0			0
$652$	0.114220	+	1.59700i	0.114220	+	1.59700i
$653$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$653$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$654$		0			0
$655$		0			0
$656$	−0.847507	+	1.13214i	−0.847507	+	1.13214i
$657$			0.864183i			0.864183i
$658$		0			0
$659$	0.153882	−	0.239446i	0.153882	−	0.239446i	−0.755750	−	0.654861i	$-0.772727\pi$
$659$	0.153882	−	0.239446i	0.153882	−	0.239446i	0.909632	+	0.415415i	$0.136364\pi$
$660$		0			0
$661$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$661$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$662$	−0.544078	−	1.19136i	−0.544078	−	1.19136i
$663$		0			0
$664$	0.415415	+	0.0903680i	0.415415	+	0.0903680i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$673$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$674$	−0.142315	−	0.0101786i	−0.142315	−	0.0101786i
$675$	−0.477540	+	0.477540i	−0.477540	+	0.477540i
$676$	0.540641	−	0.841254i	0.540641	−	0.841254i
$677$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$677$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$678$			0.509543i			0.509543i
$679$		0			0
$680$		0			0
$681$	−0.829178	−	2.22311i	−0.829178	−	2.22311i
$682$		0			0
$683$	−0.114220	−	1.59700i	−0.114220	−	1.59700i	−0.654861	−	0.755750i	$-0.727273\pi$
$683$	−0.114220	−	1.59700i	−0.114220	−	1.59700i	0.540641	−	0.841254i	$-0.318182\pi$
$684$	0.697148	−	0.521878i	0.697148	−	0.521878i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−1.54064	−	0.841254i	−1.54064	−	0.841254i
$689$		0			0
$690$		0			0
$691$	0.474017	+	1.61435i	0.474017	+	1.61435i	0.755750	+	0.654861i	$0.227273\pi$
$691$	0.474017	+	1.61435i	0.474017	+	1.61435i	−0.281733	+	0.959493i	$0.590909\pi$
$692$		0			0
$693$		0			0
$694$	−0.817178	+	0.708089i	−0.817178	+	0.708089i
$695$		0			0
$696$		0			0
$697$	1.30061	−	2.38189i	1.30061	−	2.38189i
$698$		0			0
$699$	−1.97049	+	0.428654i	−1.97049	+	0.428654i
$700$		0			0
$701$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$701$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$702$		0			0
$703$		0			0
$704$	−0.989821	+	1.14231i	−0.989821	+	1.14231i
$705$		0			0
$706$	0.334961	+	0.613435i	0.334961	+	0.613435i
$707$		0			0
$708$	−0.966443	+	0.621095i	−0.966443	+	0.621095i
$709$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$709$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$710$		0			0
$711$		0			0
$712$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.698939	−	0.449181i	0.698939	−	0.449181i
$717$		0			0
$718$		0			0
$719$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$719$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$720$		0			0
$721$		0			0
$722$	2.71038	+	1.23779i	2.71038	+	1.23779i
$723$	1.53390	+	1.77021i	1.53390	+	1.77021i
$724$		0			0
$725$		0			0
$726$	−1.44262	+	0.538070i	−1.44262	+	0.538070i
$727$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$727$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$728$		0			0
$729$	−0.0748076	+	0.254771i	−0.0748076	+	0.254771i
$730$		0			0
$731$	3.15612	+	1.17717i	3.15612	+	1.17717i
$732$		0			0
$733$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$733$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.391340	−	2.72183i	−0.391340	−	2.72183i
$738$	0.541838	−	0.295866i	0.541838	−	0.295866i
$739$	0.418852	+	0.559521i	0.418852	+	0.559521i	0.959493	−	0.281733i	$-0.0909091\pi$
$739$	0.418852	+	0.559521i	0.418852	+	0.559521i	−0.540641	+	0.841254i	$0.681818\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$743$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.148568	−	0.111217i	−0.148568	−	0.111217i
$748$	1.56815	−	2.44009i	1.56815	−	2.44009i
$749$		0			0
$750$		0			0
$751$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$751$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$752$		0			0
$753$	1.97049	+	0.428654i	1.97049	+	0.428654i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$757$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$758$	−0.133682	+	1.86912i	−0.133682	+	1.86912i
$759$		0			0
$760$		0			0
$761$	1.49611	−	0.215109i	1.49611	−	0.215109i	0.654861	−	0.755750i	$-0.272727\pi$
$761$	1.49611	−	0.215109i	1.49611	−	0.215109i	0.841254	+	0.540641i	$0.181818\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.718267	−	0.959493i	0.718267	−	0.959493i
$769$	−0.153882	+	1.07028i	−0.153882	+	1.07028i	0.755750	+	0.654861i	$0.227273\pi$
$769$	−0.153882	+	1.07028i	−0.153882	+	1.07028i	−0.909632	+	0.415415i	$0.863636\pi$
$770$		0			0
$771$	0.333679	−	1.53390i	0.333679	−	1.53390i
$772$	−0.0498610	−	0.697148i	−0.0498610	−	0.697148i
$773$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$773$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$774$	0.459211	+	0.613435i	0.459211	+	0.613435i
$775$		0			0
$776$	0.118239	+	0.822373i	0.118239	+	0.822373i
$777$		0			0
$778$		0			0
$779$	2.37336	+	1.52527i	2.37336	+	1.52527i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.281733	−	0.959493i	0.281733	−	0.959493i
$785$		0			0
$786$	−0.163493	+	0.299415i	−0.163493	+	0.299415i
$787$	−1.83107	+	0.682956i	−1.83107	+	0.682956i	−0.841254	+	0.540641i	$0.818182\pi$
$787$	−1.83107	+	0.682956i	−1.83107	+	0.682956i	−0.989821	+	0.142315i	$0.954545\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	0.600195	−	0.274100i	0.600195	−	0.274100i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$797$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$798$		0			0
$799$		0			0
$800$	1.00000			1.00000
$801$	0.122986	+	0.418852i	0.122986	+	0.418852i
$802$	−1.81926			−1.81926
$803$	−2.87102	+	0.843008i	−2.87102	+	0.843008i
$804$	0.463496	+	2.13066i	0.463496	+	2.13066i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.25667	−	1.45027i	1.25667	−	1.45027i	0.415415	−	0.909632i	$-0.363636\pi$
$809$	1.25667	−	1.45027i	1.25667	−	1.45027i	0.841254	−	0.540641i	$-0.181818\pi$
$810$		0			0
$811$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$811$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−1.10228	+	2.01867i	−1.10228	+	2.01867i
$817$	−1.45469	+	3.18532i	−1.45469	+	3.18532i
$818$	−0.234072	+	0.797176i	−0.234072	+	0.797176i
$819$		0			0
$820$		0			0
$821$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$821$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$822$	−0.323658	−	1.10228i	−0.323658	−	1.10228i
$823$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$823$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$824$		0			0
$825$	1.59002	+	0.868215i	1.59002	+	0.868215i
$826$		0			0
$827$	1.71524	−	0.936593i	1.71524	−	0.936593i	0.755750	−	0.654861i	$-0.227273\pi$
$827$	1.71524	−	0.936593i	1.71524	−	0.936593i	0.959493	−	0.281733i	$-0.0909091\pi$
$828$		0			0
$829$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$829$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.273100	+	1.89945i	−0.273100	+	1.89945i
$834$	−0.940730	+	1.25667i	−0.940730	+	1.25667i
$835$		0			0
$836$	2.41387	+	1.80700i	2.41387	+	1.80700i
$837$		0			0
$838$	−0.494217	+	0.494217i	−0.494217	+	0.494217i
$839$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$839$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$840$		0			0
$841$	−0.989821	+	0.142315i	−0.989821	+	0.142315i
$842$		0			0
$843$	−0.0924539	−	0.143861i	−0.0924539	−	0.143861i
$844$	−0.125226	+	1.75089i	−0.125226	+	1.75089i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	1.26636	+	0.275479i	1.26636	+	0.275479i
$850$	−1.89945	+	0.273100i	−1.89945	+	0.273100i
$851$		0			0
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.19550	−	1.59700i	1.19550	−	1.59700i	0.540641	−	0.841254i	$-0.318182\pi$
$857$	1.19550	−	1.59700i	1.19550	−	1.59700i	0.654861	−	0.755750i	$-0.272727\pi$
$858$		0			0
$859$	−0.148568	−	0.398326i	−0.148568	−	0.398326i	0.841254	−	0.540641i	$-0.181818\pi$
$859$	−0.148568	−	0.398326i	−0.148568	−	0.398326i	−0.989821	+	0.142315i	$0.954545\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$863$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$864$	0.592735	−	0.323658i	0.592735	−	0.323658i
$865$		0			0
$866$	−1.40524	−	0.767317i	−1.40524	−	0.767317i
$867$	1.12357	−	3.01242i	1.12357	−	3.01242i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.102181	−	0.347995i	0.102181	−	0.347995i
$874$		0			0
$875$		0			0
$876$	2.22311	−	0.829178i	2.22311	−	0.829178i
$877$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$877$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.512546	−	0.234072i	0.512546	−	0.234072i	−0.142315	−	0.989821i	$-0.545455\pi$
$881$	0.512546	−	0.234072i	0.512546	−	0.234072i	0.654861	+	0.755750i	$0.272727\pi$
$882$	−0.285870	+	0.329911i	−0.285870	+	0.329911i
$883$	1.94931	−	0.139418i	1.94931	−	0.139418i	0.959493	−	0.281733i	$-0.0909091\pi$
$883$	1.94931	−	0.139418i	1.94931	−	0.139418i	0.989821	+	0.142315i	$0.0454545\pi$
$884$		0			0
$885$		0			0
$886$	0.909632	−	0.584585i	0.909632	−	0.584585i
$887$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$887$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.88329			1.88329
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.708089	+	0.817178i	−0.708089	+	0.817178i
$899$		0			0
$900$	−0.397086	−	0.181343i	−0.397086	−	0.181343i
$901$		0			0
$902$	1.51150	+	1.51150i	1.51150	+	1.51150i
$903$		0			0
$904$	0.398326	−	0.148568i	0.398326	−	0.148568i
$905$		0			0
$906$		0			0
$907$	−0.158746	+	0.540641i	−0.158746	+	0.540641i	0.841254	+	0.540641i	$0.181818\pi$
$907$	−0.158746	+	0.540641i	−0.158746	+	0.540641i	−1.00000			$\pi$
$908$	−1.49611	+	1.29639i	−1.49611	+	1.29639i
$909$		0			0
$910$		0			0
$911$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$911$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$912$	−2.01144	−	1.29267i	−2.01144	−	1.29267i
$913$	0.224560	−	0.602069i	0.224560	−	0.602069i
$914$	1.64468	+	0.898064i	1.64468	+	0.898064i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−1.03748	+	0.776649i	−1.03748	+	0.776649i
$919$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$919$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$920$		0			0
$921$	0.704722	+	1.88943i	0.704722	+	1.88943i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.698939	−	1.53046i	−0.698939	−	1.53046i	−0.841254	−	0.540641i	$-0.818182\pi$
$929$	−0.698939	−	1.53046i	−0.698939	−	1.53046i	0.142315	−	0.989821i	$-0.454545\pi$
$930$		0			0
$931$	−1.94931	−	0.424047i	−1.94931	−	0.424047i
$932$	0.909632	+	1.41542i	0.909632	+	1.41542i
$933$		0			0
$934$	1.84125	+	0.540641i	1.84125	+	0.540641i
$935$		0			0
$936$		0			0
$937$	−0.304632	−	0.474017i	−0.304632	−	0.474017i	0.654861	−	0.755750i	$-0.272727\pi$
$937$	−0.304632	−	0.474017i	−0.304632	−	0.474017i	−0.959493	+	0.281733i	$0.909091\pi$
$938$		0			0
$939$	0.829178	−	0.119218i	0.829178	−	0.119218i
$940$		0			0
$941$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$941$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$942$		0			0
$943$		0			0
$944$	0.767317	+	0.574406i	0.767317	+	0.574406i
$945$		0			0
$946$	−1.59002	+	2.12401i	−1.59002	+	2.12401i
$947$	0.258908	−	1.80075i	0.258908	−	1.80075i	−0.281733	−	0.959493i	$-0.590909\pi$
$947$	0.258908	−	1.80075i	0.258908	−	1.80075i	0.540641	−	0.841254i	$-0.318182\pi$
$948$		0			0
$949$		0			0
$950$	−0.142315	−	1.98982i	−0.142315	−	1.98982i
$951$		0			0
$952$		0			0
$953$	1.40524	−	0.767317i	1.40524	−	0.767317i	0.415415	−	0.909632i	$-0.363636\pi$
$953$	1.40524	−	0.767317i	1.40524	−	0.767317i	0.989821	+	0.142315i	$0.0454545\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.755750	+	0.654861i	−0.755750	+	0.654861i
$962$		0			0
$963$		0			0
$964$	0.936593	−	1.71524i	0.936593	−	1.71524i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	0.841254	+	0.970858i	0.841254	+	0.970858i
$969$	4.17367	+	1.90605i	4.17367	+	1.90605i
$970$		0			0
$971$	−0.544078	+	0.627899i	−0.544078	+	0.627899i	−0.959493	−	0.281733i	$-0.909091\pi$
$971$	−0.544078	+	0.627899i	−0.544078	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$972$	−0.815938	+	0.0583571i	−0.815938	+	0.0583571i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.89945	−	0.557730i	1.89945	−	0.557730i	0.909632	−	0.415415i	$-0.136364\pi$
$977$	1.89945	−	0.557730i	1.89945	−	0.557730i	0.989821	−	0.142315i	$-0.0454545\pi$
$978$	−1.91899			−1.91899
$979$	−1.27155	+	0.817178i	−1.27155	+	0.817178i
$980$		0			0
$981$		0			0
$982$	−0.0303285	−	0.139418i	−0.0303285	−	0.139418i
$983$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$983$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$984$	−1.28101	−	1.11000i	−1.28101	−	1.11000i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$991$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$992$		0			0
$993$	1.47080	−	0.548580i	1.47080	−	0.548580i
$994$		0			0
$995$		0			0
$996$	−0.143555	+	0.488902i	−0.143555	+	0.488902i
$997$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$997$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$998$	−1.83107	−	0.682956i	−1.83107	−	0.682956i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	712.1.y.a.691.1	yes	20
4.3	odd	2		2848.1.cc.a.335.1		20
8.3	odd	2	CM	712.1.y.a.691.1	yes	20
8.5	even	2		2848.1.cc.a.335.1		20
89.72	even	44	inner	712.1.y.a.339.1	✓	20
356.339	odd	44		2848.1.cc.a.2831.1		20
712.339	odd	44	inner	712.1.y.a.339.1	✓	20
712.517	even	44		2848.1.cc.a.2831.1		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
712.1.y.a.339.1	✓	20	89.72	even	44	inner
712.1.y.a.339.1	✓	20	712.339	odd	44	inner
712.1.y.a.691.1	yes	20	1.1	even	1	trivial
712.1.y.a.691.1	yes	20	8.3	odd	2	CM
2848.1.cc.a.335.1		20	4.3	odd	2
2848.1.cc.a.335.1		20	8.5	even	2
2848.1.cc.a.2831.1		20	356.339	odd	44
2848.1.cc.a.2831.1		20	712.517	even	44

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 712 = 2^{3} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	712.y (of order \(44\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(0.959493 - 0.281733i) q^{2} +(0.254771 + 1.17116i) q^{3} +(0.841254 - 0.540641i) q^{4} +(0.574406 + 1.05195i) q^{6} +(0.654861 - 0.755750i) q^{8} +(-0.397086 + 0.181343i) q^{9} +O(q^{10})\)	\(q+(0.959493 - 0.281733i) q^{2} +(0.254771 + 1.17116i) q^{3} +(0.841254 - 0.540641i) q^{4} +(0.574406 + 1.05195i) q^{6} +(0.654861 - 0.755750i) q^{8} +(-0.397086 + 0.181343i) q^{9} +(-0.989821 - 1.14231i) q^{11} +(0.847507 + 0.847507i) q^{12} +(0.415415 - 0.909632i) q^{16} +(-0.540641 + 1.84125i) q^{17} +(-0.329911 + 0.285870i) q^{18} +(-1.86912 - 0.697148i) q^{19} +(-1.27155 - 0.817178i) q^{22} +(1.05195 + 0.574406i) q^{24} +(0.142315 + 0.989821i) q^{25} +(0.404719 + 0.540641i) q^{27} +(0.142315 - 0.989821i) q^{32} +(1.08566 - 1.45027i) q^{33} +1.91899i q^{34} +(-0.236009 + 0.367237i) q^{36} +(-1.98982 - 0.142315i) q^{38} +(-1.38189 - 0.300613i) q^{41} +(0.125226 - 1.75089i) q^{43} +(-1.45027 - 0.425839i) q^{44} +(1.17116 + 0.254771i) q^{48} +(0.989821 - 0.142315i) q^{49} +(0.415415 + 0.909632i) q^{50} +(-2.29415 - 0.164081i) q^{51} +(0.540641 + 0.404719i) q^{54} +(0.340275 - 2.36667i) q^{57} +(-0.203743 + 0.936593i) q^{59} +(-0.142315 - 0.989821i) q^{64} +(0.633095 - 1.69739i) q^{66} +(1.53046 + 0.983568i) q^{67} +(0.540641 + 1.84125i) q^{68} +(-0.122986 + 0.418852i) q^{72} +(0.822373 - 1.80075i) q^{73} +(-1.12299 + 0.418852i) q^{75} +(-1.94931 + 0.424047i) q^{76} +(-0.815938 + 0.941643i) q^{81} +(-1.41061 + 0.100889i) q^{82} +(0.203743 + 0.373128i) q^{83} +(-0.373128 - 1.71524i) q^{86} -1.51150 q^{88} +(0.142315 - 0.989821i) q^{89} +(1.19550 - 0.0855040i) q^{96} +(-0.544078 + 0.627899i) q^{97} +(0.909632 - 0.415415i) q^{98} +(0.600195 + 0.274100i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8}+O(q^{10}) \) 20 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 2 * q^8	\( 20 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8} - 2 q^{12} - 2 q^{16} - 2 q^{19} + 2 q^{24} + 2 q^{25} - 22 q^{27} + 2 q^{32} - 20 q^{38} + 2 q^{41} + 2 q^{43} - 2 q^{48} - 2 q^{50} + 4 q^{51} - 4 q^{57} - 2 q^{59} - 2 q^{64} + 22 q^{72} + 2 q^{75} - 2 q^{76} - 2 q^{81} - 2 q^{82} + 2 q^{83} - 2 q^{86} + 2 q^{89} + 2 q^{96} - 4 q^{97}+O(q^{100}) \) 20 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 2 * q^8 - 2 * q^12 - 2 * q^16 - 2 * q^19 + 2 * q^24 + 2 * q^25 - 22 * q^27 + 2 * q^32 - 20 * q^38 + 2 * q^41 + 2 * q^43 - 2 * q^48 - 2 * q^50 + 4 * q^51 - 4 * q^57 - 2 * q^59 - 2 * q^64 + 22 * q^72 + 2 * q^75 - 2 * q^76 - 2 * q^81 - 2 * q^82 + 2 * q^83 - 2 * q^86 + 2 * q^89 + 2 * q^96 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more